where \(H^0_{\mathrm {diff}}(G)\)  denotes 0th-degree groupoid cohomology, which is well-known to be Morita invariant [1]}. Hence, one can think of \(H^0_{\mathrm {diff}}(G)=C^{\infty }(M)^G\)  as the space of smooth functions on the quotient stack \([M/G]\) . On the other hand, any basic function \(f\in C^{\infty }(M)^G\)  induces a Lie groupoid morphism \(F:(G\rightrightarrows M)\rightarrow (\mathbb {R}\rightrightarrows \mathbb {R})\)  given either by \(F=s^*f\)  or \(F=t^*f\) . It is clear that every Lie groupoid morphism \(F:G\rightarrow \mathbb {R}\)  has this form. Hence one can identify the space of real valued Lie groupoids morphisms with that of basic functions.
